Respuesta :
Let us start with the explanation:
A rule of polygons says that the sum of the exterior angles always equals 360 degrees.
First of all we need to know the Interior and Exterior angle formulas:
It says the sum of the measures of the interior angles of a polygon with [tex]n[/tex] sides is [tex](n-2)\times 180=(n-2) 180[/tex].
Now If we count one exterior angle at each vertex, the sum of the measures of the exterior angles of a polygon is always 360°.
Let's talk about Hexagon first : It has got 6 sides so, [tex]n=6[/tex].
Since the skater is going around the hexagon, we need to find the sum of the measure of exterior angles ,
[tex](n-2)180=(6-2)\times 180=4\times 180=4 \times \pi= 720^ \circ=4 \pi[/tex]
Talking about Octagon, it has got 8 sides so, [tex]n=8[/tex]. So the sum of the measure of exterior angles is:
Plugging the value of 'n' we get:
[tex](8-2) \times 180=6 \times 180=1080 ^\circ=6 \pi[/tex]
Now, finding the sum of measure of exterior angles for a [tex]n[/tex] sided polygon. We get:
[tex](n-2)\times 180= (n-2) \pi[/tex]
Therefore, when the skater goes around a hexagon he covers [tex]720^\circ[/tex] , when he goes around an octagon he covers [tex]1080^\circ[/tex], and when he goes around a regular [tex]n[/tex] sided polygon , he covers [tex](n-2)\pi=(n-2)\times 180 ^\circ[/tex].
